FE450: GAME THEORY
Fall 2011 – Professor Schmitt
Homework 4 – due 17 Nov
1. (50 points) Two firms compete in selling a good; one firm does not know the other firm’s
cost function. Assume that both firms face a constant marginal cost. Assume that both
firms know that the marginal cost for firm 1 is 3.6. Firm 2 only knows its own marginal
cost, and firm 1 believes the firm 2’s marginal cost is 2.4 with probability 1 and that
3
firm 2’s marginal cost is 4.2 with probability 2 . Assume the demand curve is given
3
by: Q  240  10 P . Finally, where appropriate, assume that Firm 1 is the first-mover in
a sequential game. Assuming they are Cournot competitors, find the Nash Equilibrium
quantities.
Starting with the low cost firm 2:
 2  Pq2 Low  cq 2 Low
1
1
q1  q 2 Low )q 2 Low  2.4q 2 Low
10
10
Taking the derivative with respect to q2 Low  So the Best Response function for the low cost
1
firm 2 is given by: q2 Low = 108  q1
2
 2  (24 
For the high cost firm 2:
 2  Pq2 high  cq 2 high
 2  (24 
1
1
q1  q 2 high )q 2 high  4.2q 2 high
10
10
So the Best Response function for the high cost firm 2 is given by: q 2 high  99 
1
q1
2
Because firm 1 is not completely informed, we have to look at their profit maximizing quantities
with the given probabilities
1
3
 1  ( )( 24 
1
1
2
1
1
q1  q 2 Low )  ( )( 24  q1  q 2 high )q1  3.6q1
10
10
3
10
10
Taking the derivative with respect to q1  So the Best Response function for firm 1 is given by:
1
1
q1 = 102  q 2low  q 2 high
6
3
Simultaneously solving:
q2 high = 65; q2low = 74; q1 = 68
If you did stackelberg, my apologies, the answer is below:
1
3
 1stackelberg  ( )( 24 
1
2
1
(q1  q 2 Low )) q1  ( )( 24  (q1  q 2 high )q1  3.6q1
10
3
10
Substitution of the BR for high and lost cost firm 2:
1
3
q2 Low
1
1
2
1
1
(q1  (108  q1 ))) q1  ( )( 24  (q1  (99  q1 )) q1  3.6q1
10
2
3
10
2
=57; q1 =102
 1stackelberg  ( )( 24 
q 2 high 
48;
2. (50 points) Sue (player 1) and Bill (player 2) are deciding to go out Friday night. Their
options for Friday night entertainment are “dinner and a movie” and “going to a club”.
Assume there are two types for each player. That is, Sue can either “like” or “avoid” Bill
and Bill can either “like” or “avoid” Sue. Sue knows her own type and Bill knows that
with probability 2/3 Sue “likes” Bill. Bill knows his own type and Sue knows that with
probability ½ Bill “likes” Sue. Assume the following matrices for each type:
Type (like) Bill prob ½
Bill
Dinner
Club
Sue
Dinner
2, 1
0, 0
Club
0, 0
1, 2
Type: like – Sue prob 2/3 (TOP)
Sue
Type: like – Bill prob ½
Sue
Bill
Dinner
Dinner
0, 1
Club
1, 0
Club
2, 0
0, 2
Type (like) Bill prob ½
Bill
Dinner
Club
Dinner
2, 0
0, 2
Club
0, 1
1, 0
Type: avoid Bill prob ½
Sue
Bill
Dinner
Dinner
0, 0
Club
1, 1
Club
2, 2
0, 0
Type: avoid Sue prob 1/3 (BOTTOM)
Identify the pure strategy Nash Equilibria, if any exist. Be sure to show all the work).
Matrix is now:
DLDA
DLCA
CLDA
CLCA
DLDA
(2,0) (1,0)
(2,1) (2/3, 1/3)
(0,0) (1/3, 2/3)
(0,1) (0,1)
DLCA
(1,1) (1,2)
(1, ½ )(2/3, 4/3)
( ½ ,1)(1/3, 1/3)
( ½ , ½ )(0,0)
NE: {( DLDA),( DLCA)}, {( CLDA),( CLCA)}
CLDA
(1,1) (0, 0)
(1, ½ ) (2/3, 1/3)
( ½ ,1)(4/3, 2/3)
( ½ , ½ ) (2,1)
CLCA
(0,2) (0,2)
(0,0) (2/3, 4/3)
(1,2) (4/3, 2/3)
(1,0) (2,0)